A solution to Erd\H{o}s and Hajnal's odd cycle problem

TL;DR

This paper resolves Erd ext{"o}s and Hajnal's odd cycle problem by establishing a lower bound on the sum of reciprocals of odd cycle lengths in graphs with high chromatic number, and applies similar methods to solve related problems about cycles and subdivisions.

Contribution

It provides the first asymptotically optimal bound for the sum of reciprocals of odd cycle lengths in high chromatic graphs and extends techniques to cycle length and subdivision problems.

Findings

Sum of reciprocals of odd cycle lengths grows with log of chromatic number

Average degree condition guarantees cycles with lengths as powers of 2

High average degree ensures subdivisions of complete graphs with uniform edge subdivisions

Abstract

In 1981, Erd\H{o}s and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let $\mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let $\mathcal{C}_\text{odd}(G)$ be the set of odd numbers in $\mathcal{C}(G)$. We prove that, if $G$ has chromatic number $k$, then $\sum_{\ell\in \mathcal{C}_\text{odd}(G)}1/\ell\geq (1/2-o_k(1))\log k$. This solves Erd\H{o}s and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erd\H{o}s asked whether there is some $d$ such that each graph with chromatic number at least $d$ (or perhaps even only average degree at least $d$) has a cycle whose length is a power of 2. We show that an average degree condition is sufficient for this problem, solving it with methods that apply to a wide range of sequences in addition to the powers of 2. Finally, we use our methods to show that, for every $k$, there is some $d$ so that every graph with average degree at least $d$ has a subdivision of the complete graph $K_k$ in which each edge is subdivided the same number of times. This confirms a conjecture of Thomassen from 1984.